ccxxii My God, He Plays Dice! David Bohm s Hidden Variables
Hidden Variablesccxxiii David Bohm s Hidden Variables David Bohm is perhaps best known for new experimental methods to test Einstein s supposed suggestion of hidden variables that would explain the EPR paradox by providing the information needed at the distant entangled particle, so it can coordinate its properties perfectly with the local particle. Bohm wrote in 1952, The usual interpretation of the quantum theory is based on an assumption having very far-reaching implications, viz., that the physical state of an individual system is completely specified by a wave function that determines only the probabilities of actual results that can be obtained in a statistical ensemble of similar experiments. This assumption has been the object of severe criticisms, notably on the part of Einstein, who has always believed that, even at the quantum level, there must exist precisely definable elements or dynamical variables determining (as in classical physics) the actual behavior of each individual system, and not merely its probable behavior. Since these elements or variables are not now included in the quantum theory and have not yet been detected experimentally, Einstein has always regarded the present form of the quantum theory as incomplete, although he admits its internal consistency. 1 Five years later, Bohm and his Israeli student Yakir Aharonov reformulated the original EPR argument in terms of electron spin. They said experimental tests with continuous variables would be much more difficult than tests with discrete quantities, such as the spin of electrons or polarization of photons. They wrote: We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore ψ = (1/ 2) [ ψ+ (1) ψ- (2) - ψ- (1) ψ+ (2) ] where ψ+ (1) refers to the wave function of the atomic state in which one particle (A) has spin +ħ/2, etc. The two atoms are then separated by a method that does not influence the total 1 Bohm 1952, p.166
ccxxiv My God, He Plays Dice! spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle (A) is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A. 2 Einstein may have pressed Bohm to develop hidden variables as the source of nonlocal behavior. Einstein had heartily approved of Bohm s textbook and was initially supportive of Bohm s new mechanics. Einstein thought Bohm was young enough and smart enough to produce the mathematical arguments that the older generation of determinist physicists like Erwin Schrödinger, Max Planck, and others had not been able to accomplish. But when Bohm finished the work, based on Louis de Broglie s 1923 pilot-wave idea (which Einstein had supported), Einstein rejected it, as he always had rejected nonlocality, as inconsistent with his theory of relativity. No Hidden Variables, but Hidden Constants? There may be no hidden variables, local or nonlocal. But there are hidden constants. Hidden in plain sight, they are the constants of the motion, conserved quantities like energy, momentum, angular momentum, and spin, both electron and photon. As Bohm and Aharonov say above, because the total spin is still zero,...the spin of the other particle (B) is opposite to that of A. Einstein s objective reality asks whether the particles might not simply have continuous paths from the start of the experiment to the final measurement(s), although the limits of quantum measurement may never allow us to know those paths. Conservation of momentum requires that positions where they finally do appear are equidistant from the origin, in order to conserve linear momentum. And every other conserved quantity, like angular momentum, electron or photon spin, as well as energy, also appear perfectly correlated at the two symmetric positions. It is the fundamental principle of conservation that governs the correlated outcome, not some hypothetical, faster than light, communication of information between the particles. 2 Bohm and Aharonov, 1957, p. 1070
Hidden Variablesccxxv Einstein s objective reality suggests that the conservation laws hold at every position along the path from the preparation of the entangled particles to their measurement. Just because we cannot measure positions and paths does not mean that they don t exist. The hidden constants of the motion then include the electron spins suggested by Bohm as the best test for the hidden variables needed to support nonseparability and entanglement. If the two particles have had the same opposing spins since thir state preparation at the start of the experiment, they will have the perfectly correlated opposing spins when they get measured. No information need be transferred between the particles. The particles carry the needed information with them as they follow Einstein s objective paths. Bohmian Mechanics Bohm is also well known for his Bohmian Mechanics, a formulation of non-relativistic quantum mechanics that emphasizes the motion of particles and promises to restore causality to physics. It is a deterministic theory, one of several interpretations that are the most popular alternatives to the Copenhagen Interpretation. By emphasizing the motion of particles, Bohmian mechanics somewhat deemphasizes the wave function Ψ, limiting its role to guiding the motion of the particles, in comparison to competitive interpretations that deny the existence of particles altogether. Bohmian mechanics includes a mechanism whereby physical effects can move faster than light, providing an explanation for Einstein s nonlocality. But as we saw above, Einstein s objective reality provides a simpler solution that removes any conflict between relativity and quantum mechanics, Bohmian mechanics describes particles as moving along continuous paths, agreeing with Einstein s objective reality. In the famous two-slit experiment, Bohm s particles always move through just one slit, even as the guiding wave function moves through both when both are open. This provides an intuitive
ccxxvi My God, He Plays Dice! solution to the two-slit experiment, which Richard Feynman calls the one mystery in quantum mechanics. See chapter 33. The Bohmian solution involves three simple steps First, close slit 1 and open slit 2. The particle goes through slit 2. It arrives at x on the plate with probability ψ2(x) 2, where ψ2 is the wave function which passed through slit 2. Second, close slit 2 and open slit 1. The particle goes through slit 1. It arrives at x on the plate with probability ψ1(x) 2, where ψ1 is the wave function which passed through slit 1. Third, open both slits. The particle goes through slit 1 or slit 2. It arrives at x with probability ψ1(x)+ψ2(x) 2. Now observe that in general, ψ1(x)+ψ2(x) 2 = ψ1(x)+ψ2(x) 2= ψ1(x) 2+ ψ2(x) 2 + 2Rψ*1(x) ψ2(x). The last term comes from the interference of the wave packets ψ1 and ψ2 which passed through slit 1 and slit 2. The probabilities of finding particles when both slits are open are different from the sum of slit 1 open and slit 2 open separately. The wave function determines the probabilities of finding particles, just as Einstein first proposed. 3 Irreversibility In his excellent 1951 textbook, Quantum Theory, Bohm described the necessity for irreversibility in any measurement. Bohm followed John von Neumann s measurement theory in which recorded data is irreversible. A measurement has only been made when new 3 Dürr and Teufel, 2009, p.9
Hidden Variables ccxxvii information has come into the world and adequate entropy has been carried away to insure the stability of the new information, long enough for it to be observed by the conscious observer. From the previous work it follows that a measurement process is irreversible in the sense that, after it has occurred, re-establishment of definite phase relations between the eigenfunctions of the measured variable is overwhelmingly unlikely. This irreversibility greatly resembles that which appears in thermodynamic processes, where a decrease of entropy is also an overwhelmingly unlikely possibility... Because the irreversible behavior of the measuring apparatus is essential for the destruction of definite phase relations and because, in turn, the destruction of definite phase relations is essential for the consistency of the quantum theory as a whole, it follows that thermodynamic irreversibility enters into the quantum theory in an integral way. This is in remarkable contrast to classical theory, where the concept of thermoynamic irreversibility plays no fundamental role in the basic sciences of mechanics and electrodynamics. 4 But Bohm changed his mind when he developed his more realistic and deterministic theory. Now he became concerned with the problem of microscopic irreversibility. See chapter 12. 4 Bohm, 1951, p.168
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